Question

What is the slope of a line perpendicular to the x-axis?

Answer 1

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Explanation:

the slope of a line parallel to the `x`-axis has slope `0`.

the slope of a line perpendicular to another will have a slope which is its negative reciprocal.

the negative reciprocal of a number is `-1` divided by the number (e.g. the negative reciprocal of `2` is `(-1)/2`, which is `-1/2`).

the negative reciprocal of `0` is `-1/0`.

this is undefined, since one cannot define the value of any number that is divided by `0`.

Answer 2

We say vertical lines have "no slope," horizontal lines have zero slope. The equation is `x=text{constant}` so it's not equivalent to any slope-intercept form `y=mx+b.` The slope is undefined because the denominator, change in `x`, is zero.

Explanation:

One may use a direction vector, `(p,q),` instead of a slope. It's equivalent to a slope `q/p` but works when `p=0.` A line is expressed in parametric form: `(x,y)=(a,b)+t(p,q)` where `t` ranges over the reals. The parameter `t` forms a natural ruler along the line, each increment of one in `t` is a length `sqrt{p^2+q^2}` along the line.